rotation and translation matrix from homography opencv

Question

我正在Java中的android应用程序上工作，我使用findHomography（）用opencv计算了单应性，我如何找到包含通过单应性旋转和平移的相机姿势矩阵

 Mat homography = Calib3d.findHomography(ReferencePoints2, ReferencePoints1,0,Calib3d.RANSAC);

Answer 1

You need to know the intrinsic parameters of the camera to do that.

Considers the z=0 plane. The point

 X=(x,y,0,1)'

is projected to the image as

p=P*X.

Now use the decomposition

P=K[R t],

where K is the calibration matrix and [R t] are extrinsic parameters. Since z=0, the third column vector of R is multiplied by zero. We can now drop the 3rd column to get

p=K*[r1 r2 t]*(x,y,1)=H*(x,y,1),

where H is a planar homography.

You have already computed H from eg known points. The first and second column of R and the vector t can now be recovered

[r1 r2 t]=inv(K)*H.

Make sure that r1 and r2 are unit length, then t is the correct translation vector. The third column vector of R can be recovered because R is orthogonal, for example using the cross product.

r3=cross(r1,r2).

Since H is a measurement, the r1 and r2 you computed are not exact. You can use the SVD for obtaining the closest rotation matrix to a measurement. You can then compose a projection matrix

P=K[r1 r2 r3 t]

which projects any 3D point in the coordinate frame based on your 2D coordinate system of the homograohy.

Here is some course material, which describes this situation.

https://www.dropbox.com/s/qkulg4j64lyn0qa/2018_proj_geo_for_cv_projcv_assignment.pdf?dl=0

Here is a related question. Computing camera pose with homography matrix based on 4 coplanar points

As @nbsrujan (thanks) pointed out, for those using OpenCV, there is a function which can decompose a homography into translation and rotation matrices given the intrinsics.

Answer 2

OpenCV has a function which can decompose Homography to translation and rotation matrices. But, we have to choose correct translation and rotation matrix pair from array of possible matrices returned by this function.

normal vector is surface normal in first camera's frame. If you know camera's rotation in the world for initial frame, it can be used to filter out correct translation and rotation pair from list of possibilities.